A manufacturing company produces a batch of 10 containers out of which 4 are…
2023
A manufacturing company produces a batch of 10 containers out of which 4 are defective. In a quality test, 3 containers are chosen at random. What is the probability that at least one is defective?
- A.
1/4
- B.
2/3
- C.
5/6
- D.
2/9
Show answer & explanation
Correct answer: C
For ‘at least one’ probability questions in selection-without-replacement problems, it is usually easier to use the complement rule: P(at least one favourable event) = 1 − P(none of the favourable event). This avoids separately adding the probabilities of exactly one, exactly two, and exactly three occurrences.
The batch has 10 containers in total: 4 defective and 6 non-defective. 3 containers are chosen at random.
Total number of ways to choose 3 containers out of 10: C(10,3) = 120.
Number of ways to choose 3 containers with none defective (all 3 from the 6 non-defective containers): C(6,3) = 20.
Probability that none of the 3 chosen containers is defective: 20/120 = 1/6.
By the complement rule, probability that at least one chosen container is defective: 1 − 1/6 = 5/6.
Cross-check by summing the probabilities of exactly one, exactly two, and exactly three defective containers: [C(4,1)×C(6,2) + C(4,2)×C(6,1) + C(4,3)×C(6,0)] / C(10,3) = (60 + 36 + 4) / 120 = 100/120 = 5/6. This matches the complement result of 5/6, confirming the answer.
Therefore, the probability that at least one of the 3 chosen containers is defective is 5/6.